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Unformatted text preview: Chapter 2 Inference In statistics, we typically want to use sample data to make statements about the population from which it was drawn. We call this process statistical inference. The most basic form of inference is simply the use of a sample statistic, like the sample mean, to estimate a corresponding population pa- rameter, like the population mean. But we can also use more sophisticated types of inference to make more useful statements about the population. We can classify these types of inference into two broad categories: hypothesis tests confidence intervals Well review what these are, how we do them, and how we interpret the results. This review will correspond mainly to Chapters 8 and 9 in the textbook. (Chapter 10 extends the ideas of Chapters 8 and 9 to different kinds of situations.) 2.1 Hypothesis Tests In a hypothesis test, also called a significance test, we give two possible state- ments about the population, and we see which statement the data supports. Often we phrase these statements in terms of some population parameter. Different types of data call for different types of hypothesis tests, but all hypothesis tests work in the same basic way. 2.1 Hypothesis Tests 18 The Five Steps of a Hypothesis Test A hypothesis test consists of five steps: 1. Check any relevant assumptions. 2. State the two hypotheses. 3. Calculate the test statistic. 4. Determine the p-value. 5. Make a decision. The exact procedure for a particular step will vary depending on what type of hypothesis test were doing, but every test will include all the steps. Lets go through them one by one. Assumptions Typically a hypothesis test will make assumptions about the nature of the data (for example, that it comes from a or that a particular variable has a normal distribution). We should check these assumptions as best we can before we start in order to make sure that our results will be reliable. Well talk more later about what we mean by reliable. Hypotheses A hypothesis test compares two hypotheses: The null hypothesis , H , usually represents the idea of or the idea that we intend to show evidence against. Often H states that a parameter is equal to some value. The alternative hypothesis , H a , usually represents the or the idea that we intend to show evidence for. When H states that a parameter is equal to some value, H a states that the parameter is greater than or less than that value (one-sided), or simply not equal to that value (two-sided). 2.1 Hypothesis Tests 19 The test works by pretending that H is true and checking whether the observed data is reasonable under H . If not, then we decide to believe H a instead. Thus the burden of proof is on the alternative hypothesis, in the sense that the test works by believing the the null hypothesis unless the data convinces us otherwise....
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